A Level Maths: Functions Topic Summary and Resources
Video Lessons
Watch alongside the worksheet for the full lesson experience, then test your understanding with the lesson questions.
Revision Notes
Handwritten notes summarising the key ideas for each lesson. Ideal for quick review before a test.
Exam Questions
Past-paper-style questions organised by topic, with full mark schemes.
- Functions (Domain, Range and Inverse) (OCR)
- Functions (Domain, Range and Inverse) (Edexcel)
- The Modulus Function (OCR)
- The Modulus Function (Edexcel)
- Multiple Transformations Of Functions (OCR)
- Multiple Transformations Of Functions (Edexcel)
Drawn from OCR and Edexcel past papers but designed to be useful for students of all UK exam boards — including AQA and OCR MEI — unless a sheet is explicitly board-specific.
Before You Start This Topic
It will help if you are confident with the following:
- Transformations — needed for sketching transformed function graphs
- Quadratic Equations — needed for solving and rearranging
- Indices — needed for handling powers in domain restrictions
A Level Maths functions is a Year 2 Pure topic where you formalise the idea of a function as a mapping, learn to find inverses and compositions, and work carefully with domains and ranges. This is conceptually important and underpins The Modulus Function, Trigonometry: Small Angle Approximations and Inverse Trig Functions, and many calculus topics.
You distinguish between one-one and many-one mappings, and understand that only one-one functions have inverses. You find the domain (allowed inputs) and range (resulting outputs) of a function — paying close attention to natural restrictions (like $x > 0$ for $\log(x)$, $x \ne 0$ for $1/x$, and the limits imposed by square roots). You find inverse functions by swapping $x$ and $y$ and rearranging, and you understand that the graph of $y = f^{-1}(x)$ is the reflection of $y = f(x)$ in the line $y = x$. You build composite functions like $fg(x)$, meaning 'do $g$ first, then $f$', and find the domain of the composite (which can be more restrictive than either function alone). You also apply combinations of Transformations to functions, sketching the resulting graphs.
Functions is part of the Pure Maths strand of A Level Maths for AQA, Edexcel, OCR, and OCR MEI students.
Watch out for…
A few things to be careful with: the order matters in composite functions — $fg(x)$ means apply $g$ FIRST, then $f$; an inverse function exists only when the original function is one-one — you may need to restrict the domain to ensure this; the range of $f$ is the domain of $f^{-1}$ and vice versa; and when finding the domain of a composite, both functions must be defined at the appropriate points — check that the output of the inner function lies in the domain of the outer.